On the Planck-Einstein Relation Peter Langdon Ward United States Geological Survey Retired, Science Is Never Settled, Inc., Jackson, Wyoming, USA Email address: [email protected] To cite this article: Peter Langdon Ward. On the Planck-Einstein Relation. Received: MM DD, 2020; Accepted: MM DD, 2020; Published: MM DD, 2020 Abstract: The Planck-Einstein relation (E=hν) is a formula integral to quantum mechanics that says a quantum of energy (E), commonly thought of as a photon, is equal to the Planck constant (h) times a frequency of oscillation of an atomic oscillator (ν, the Greek letter nu). Yet frequency (ν) is not quantized—frequency of electromagnetic radiation is well known in Nature to be a continuum extending over at least 18 orders of magnitude from extremely low frequency, low-energy radio signals to extremely high-frequency, high-energy gamma rays. Therefore, electromagnetic energy (E), which simply equals a scaling constant (h) times a continuum (ν), must also be a continuum. We must conclude, therefore, that electromagnetic energy is not quantized at the submicroscopic level as widely assumed. Secondly, it makes no physical sense in Nature to add frequencies of electromagnetic radiation together in air or space—red light plus blue light does not physically equal ultraviolet light. Therefore, if E=hν, then it makes no physical sense to add together electromagnetic energies that are commonly thought of as photons. The purpose of this paper is to look at the history of E=hν and to examine the implications of accepting E=hν as a valid description of physical reality. Recognizing the role of E=hν makes much of the fundamental physics studied in quantum mechanics both physically intuitive and deterministic. Keywords: Planck-Einstein Relation, Planck’s Law, Photon, Atomic Oscillator, Dissociation, Resonance, Continuum, Electromagnetic Radiation 1. Introduction On Sunday, October 7, 1900, Heinrich Rubens and wife visited Max Planck and wife for afternoon tea [1]. During the social conversation, Rubens mentioned that his most recent experimental data measuring spectral radiance from a black body at the longest wavelengths yet available, fit a new law suggested by Lord Rayleigh [2] better than the empirical distribution law proposed by Wien [3], for which Planck had worked out a derivation. By dinner, Planck sent a postcard to Rubens with a slight modification to Wien’s law, blending the two laws together in the extremes by adding a minus one after the exponential term in the denominator (Figure 1). Within two days, Rubens informed Planck that Planck’s new law for the spectral radiance of electromagnetic radiation from a black Figure 1. Planck’s law shows that radiation from a warmer body has higher body at thermal equilibrium fit all available data extremely spectral radiance (spectral amplitudes of oscillation) at all frequencies than well. does radiation from a cooler body and exhibits its greatest amplitude at a higher frequency. Each solid line shows the spectral radiance radiated from a For Planck, however, this new equation was just “lucky body at thermal equilibrium for the temperature shown. The equation for intuition” stumbled on late that Sunday afternoon. As a Planck’s law is shown in the upper left where Bν is the spectral radiance theoretical physicist, he felt driven to derive the new equation written in terms of Planck’s constant (h), frequency (ν), velocity of light (c), from first principles based on Maxwell’s wave theory of the Boltzmann constant (kB), and absolute temperature (T). 2 Peter Langdon Ward: On the Planck-Einstein Relation electromagnetic radiation. He explained the new equation at that can describe one physical phenomenon. But which of the German Physical Society on October 19 and "after some these systems describes most accurately what is actually weeks of the most strenuous work of my life" [4], presented a happening physically? derivation at the Society on 14 December. Planck [5] shows In 1865, Maxwell [12] introduced a mathematics of electro- by reformulating Wien’s displacement law, applying it to magnetic radiation assuming it propagates as waves. In 1900, Planck’s equation for entropy, and assuming that radiant Planck [5] stumbled empirically onto an equation that energy consists of an integral number of “energy elements,” predicted observed values of radiance precisely and worked “we then find that the energy element ε must be proportional very hard to derive this equation from Maxwell’s mathematics to the frequency ν, thus: ε=hν.” rather than trying to understand what was physically Einstein [6], in 1905, was the first scientist to take Planck’s happening. In so doing, he had to assume E=hν, but he never “energy elements” seriously, proposing what Einstein called a stopped to think what this simple equation actually means “light quantum” (E=hν) to explain the photoelectric effect and physically. He was obsessed with finding a mathematical then arguing that light itself is quantized. Robert Millikan was derivation based on Maxwell’s wave theory. In 1905, Einstein convinced that Einstein’s particle theory of light must be [6] then initiated a very different system of mathematics of wrong because of overwhelming support for Maxwell’s wave EMR based on the photoelectric effect where he might have theory of light, but after nearly a decade of laboratory work, assumed that an electron could be knocked loose by a quantum Millikan [7] confirmed Einstein’s predictions in every detail. of light, billiard-ball style, although I have not found him using Compton [8] explained the Compton effect in terms of those words. “radiation quanta” colliding with electrons, and de Broglie [9] Ever since, experiments in physics have been designed, suggested that particles have a wave-like nature and, therefore, interpreted, and documented assuming that EMR travels would have a wavelength. Lewis [10] call this “light quantum” through space either as waves or as photons [13]. Both systems a photon as wave-particle duality for electromagnetic radiation of mathematics have become highly developed, building on (EMR) began to be widely accepted. E=hν became known as their initial assumptions, and trying to fit each theory to the Planck-Einstein relation where E is widely accepted today increasingly complex experimental data. as the energy of a photon. Most physicists today also think that Physicists to this day, think in terms of wave-particle duality, the energies of photons can be added together to provide a where they utilize whichever mathematics seems most greater amount of energy—i.e. energy at the submicroscopic appropriate to solve the problem at hand. The only two options level is currently thought of as both quantized and additive. on the table have been waves or particles. In terms of basic Planck [5] considered E=hν to be a “mathematical trick or logic, if you have a piece of fruit that is sort of like a banana convenience” and never appears to have thought deeply about (wave) and sort of like an orange (particle), it is clearly not its physical meaning. In 1931, he explained that introducing equal to either. the energy quantum in 1900 was “a purely formal assumption Thermal energy in matter is well known to be a broad and I really did not give it much thought except that no matter spectrum or continuum of frequencies of oscillation of all the what the cost, I must bring about a positive result” deriving bonds that hold matter together (Figure 2). It is the motion of what is now known as Planck’s law (Figure 1) from Maxwell’s charge caused by these oscillations on the surface of matter wave theory of light [11]. If Planck had thought more deeply that appears to induce EMR. What is most surprising is that I about E=hν, he might have noticed that it says energy lies on cannot find anyone, in the history of physics, who seriously an alternative x-axis, shown along the top of Figure 1, but considered EMR simply to be a broad spectrum of frequencies spectral radiance on the y-axis, as plotted, is also a function of of oscillation. Most physicists think of frequency as wave energy—something that is not quite right, as discussed below. frequency and have trouble thinking of frequency as a physical Planck was too wrapped up in the mathematics, to think carefully about the physics. 2. Excellent Mathematics Does Not Require Good Physics Physics is the study of Nature, how matter and radiation behave, move, and interact throughout space and time. Mathematics, on the other hand, is logical deductive reasoning based on initial assumptions. Good mathematics is required to do excellent physics, but good physics is not required to do excellent mathematics. Excellent mathematics can be built on assumptions that may or may not be what is actually happening physically. Excellent mathematics can take you in logical directions that are not physically possible even though Figure 2. The electromagnetic spectrum extends from radio frequencies of the mathematics is done in a rigorous and internally consistent cycles per second to gamma rays with frequencies greater than 300 Exahertz (3x1020). Oscillators with a shorter length emit higher frequency and higher way. There are often many different systems of mathematics energy radiation associated with higher temperatures. 3 thing. Frequency is invisible, but it is the result of oscillation is not quantized. of molecular bonds. Thinking about EMR, experimenting with E=hν does appear to be a valid description of physical EMR, expanding equations describing EMR have all been reality. Every description of the electromagnetic spectrum uses based on the initial assumption, that electromagnetic radiation this equation to show how energy increases with increasing travels as either a wave, a particle, or both. These initial frequency from radio signals to microwaves, to infrared assumptions do not appear to have been seriously questioned.